Probabilistic Measure of Symmetry Stability
Abstract
1. Introduction
2. Results
2.1. Introducing the Probabilistic Measure of the Symmetry Stability
2.2. Calculation of the Probabilistic Measure of the Symmetry Stability for 2p Equidistant Points Placed on the Same Straight Line
- (i)
- . Removing a single point leaves its partner unpaired, so no longer maps the remaining set to itself. Thus, we calculate
- (ii)
- . There are unordered pairs of points that could be removed. Exactly p of those pairs are mirror-pairs (one per mirror-pair). If the removed pair is one of these p pairs, the reflection survives; otherwise, symmetry is lost. So, the probability that the removed pair is a mirror-pair is given by Equation (4):
2.3. Calculation of the Probabilistic Measure of the Symmetry Stability for Symmetrical Triangles
2.4. Calculation of the Probabilistic Measure of the Symmetry Stability for the Sets Built of Four Points
2.5. Calculation of the Probabilistic Measure of the Symmetry Stability for Regular Polygons
2.6. Calculation of the Probabilistic Measure of the Symmetry Stability for Tetrahedron and Octahedron
2.7. Calculation of the Probabilistic Measure of the Symmetry Stability for Cubic Crystallographic Cells
2.8. Shannon Probabilistic Measure of the Symmetry Stability
3. Discussion
- (i)
- Extension to higher-dimensional and complex point sets: While we considered 2D polygons and 3D polyhedra, many systems of interest—such as quasicrystals, complex molecular clusters, and high-dimensional lattices—pose challenging combinatorial problems. Extending calculations to these structures could uncover novel symmetry robustness patterns.
- (ii)
- Study of probabilistic symmetry in dynamic systems: In physical and biological systems, perturbations often occur continuously rather than as discrete deletions. Developing a time-dependent or stochastic version of could quantify the resilience of symmetry under fluctuating forces, thermal noise, or dynamic defects. Study of the time evolution of is of particular interest.
- (iii)
- Connection with statistical physics and phase transitions:
- (iv)
- Systematic computation of for various crystal lattices (including HCP and more complicated structures) can inform defect-tolerance studies, mechanical stability, and design of robust nanostructures. The probabilistic framework could guide the development of materials resistant to random defects.
- (v)
- Integration with network theory and combinatorics looks attractive. Symmetry stability can be generalized to networks with geometric embedding, where nodes or edges are removed randomly. This opens potential connections with probabilistic graph theory, random automorphism groups, and combinatorial optimization.
- (vi)
- Algorithmic and computational development is instructive. Efficient algorithms for the exact or approximate computation of the introduced and the Shannon symmetry entropy Sh for large or high-symmetry point sets will be crucial. Monte Carlo simulations, group-theoretic enumeration, and probabilistic combinatorial techniques can all play a role.
- (vii)
- Experimental validation of the suggested ideas is desirable. Measuring symmetry survival probabilities in real physical systems—such as nanoparticles under random vacancy formation, molecules with isotopic substitutions, or lattice defects under irradiation—could validate and calibrate the theoretical framework, bridging theory with experimental observation.
4. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Calculation of the Probabilistic Measure of the Symmetry Stability for the Directed Necklace of Points
- (i)
- p is prime. For all nontrivial rotations, . Hence, for , every nontrivial rotation destroys the symmetry, and, consequently, .
- (ii)
- p is composite. Some rotations have orbits. Then, it is possible that is an integer, in which case the removal may leave a nontrivial symmetry intact. The exact probability that the remaining set has a nontrivial rotational symmetry can be computed via the Möbius-function formula:
Appendix B. Calculation of the Probabilistic Measure of the Symmetry Stability for Octahedron
- (i)
- Let us remove a single vertex , say five vertices remain. The remaining set contains its opposite and the two opposite pairs , . There is at least the reflection in the plane (or rotation about the x-axis composed with other symmetries of the octahedral group) that preserves the remaining five points while not being the identity. Hence, the remaining five-point set has a nontrivial symmetry.
- (ii)
- Remove two vertices (four vertices remain). There are two cases for the two removed vertices:
- (a)
- They are an opposite pair (e.g., removed). The remaining four are the two opposite pairs.that set is a square in the plane , dihedral -type nontrivial symmetry.
- (b)
- They are from different opposite pairs (e.g., , and removed). Then, the remaining set contains the whole opposite pair . rotation about the z-axis (or the reflection in the plane ) preserves the set and is nontrivial. In every subcase, at least one nontrivial isometry of the octahedral group preserves the remaining four points, so
- (iii)
- Remove three vertices three vertices remain.Up to symmetry, the only essentially different possibility is removing one vertex from each opposite pair (if we remove both members of some pair you reduce to the one of the types above). For example, remove , and ; the remaining three points are , and . These vertices are located at the vertices of an equilateral triangle (distances equal) and are permuted cyclically by a —rotation (a coordinate permutation); hence, their point-set symmetry group contains a nontrivial rotation ( subgroup). We conclude that any three-vertex remainder has nontrivial symmetry.
- (iv)
- Remove four vertices ; two vertices remain.Any two-point set has a nontrivial isometry that swaps the two points (reflection in the perpendicular bisector plane or rotation about an appropriate axis). So, the remaining pair always has a nontrivial symmetry.
- (v)
- Remove five vertices ; one vertex remains.A single point has an infinite stabilizer (all rotations/reflections fixing that point), so the symmetry is certainly nontrivial.
References
- Finnerty, J.H. The origins of axial patterning in the metazoa: How old is bilateral symmetry? Int. J. Dev. Biol. 2003, 47, 523–529. [Google Scholar] [PubMed]
- Finnerty, J.H.; Pang, K.; Burton, P.; Paulson, D.; Martindale, M.Q. Origins of Bilateral Symmetry: Hox and Dpp Expression in a Sea Anemone. Science 2004, 304, 1335. [Google Scholar] [CrossRef] [PubMed]
- Longo, G.; Montévil, M. From Physics to Biology by Extending Criticality and Symmetry Breakings. In Perspectives on Organisms; Lecture Notes in Morphogenesis; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Yonekura, T.; Sugiyama, M. Symmetry and its transition in phyllotaxis. J. Plant. Res. 2021, 134, 417–430. [Google Scholar] [CrossRef] [PubMed]
- Dumais, J. Can mechanics control pattern formation in plants? Curr. Opin. Plant Biol. 2007, 10, 58–62. [Google Scholar] [CrossRef]
- Dengler, N.G. Anisophylly and dorsiventral shoot symmetry. Int. J. Plant Sci. 1999, 160, S67–S80. [Google Scholar] [CrossRef]
- Cannon, K.A.; Ochoa, J.M.; Yeates, T.O. High-symmetry protein assemblies: Patterns and emerging applications. Curr. Opin. Struct. Biol. 2019, 55, 77–84. [Google Scholar] [CrossRef]
- Wolynes, P.G. Symmetry and the energy landscapes of biomolecules. Proc. Natl. Acad. Sci. USA 1996, 93, 14249–14255. [Google Scholar] [CrossRef]
- Hollo, G. A new paradigm for animal symmetry. Interface Focus 2015, 5, 20150032. [Google Scholar] [CrossRef]
- Bormashenko, E. Fibonacci Sequences, Symmetry and Order in Biological Patterns, Their Sources, Information Origin and the Landauer Principle. Biophysica 2022, 2, 292–307. [Google Scholar] [CrossRef]
- Weyl, H. Symmetry; Princeton University Press: Princeton, NJ, USA, 1989. [Google Scholar]
- Schwichtenberg, J. Physics from Symmetry, 2nd ed.; Springer International Publishing AG: Cham, Switzerland, 2018. [Google Scholar]
- McGreevy, J. Generalized Symmetries in Condensed Matter. Ann. Rev. Condens. Matter Phys. 2023, 14, 57–82. [Google Scholar] [CrossRef]
- Jaffé, H.H.; Orchin, M. Symmetry in Chemistry; John Wiley & Sons: New York, NY, USA, 2002. [Google Scholar]
- Hargittai, M.; Hargittai, I. Symmetry in Chemistry. Eur. Rev. 2005, 13 (Suppl. S2), 61–75. [Google Scholar] [CrossRef]
- Müller, U.; De La Flor, G. Symmetry Relationships Between Crystal Structures: Applications of Crystallographic Group Theory in Crystal Chemistry; Oxford University Press: Oxford, UK, 2024. [Google Scholar]
- Nespolo, M.; Benahsene, A.H. Symmetry and chirality in crystals. J. Appl. Cryst. 2021, 54, 1594–1599. [Google Scholar] [CrossRef]
- Putkaradze, V. Noether’s Theorem and Conservation Laws. In A Concise Introduction to Classical Mechanics. Surveys and Tutorials in the Applied Mathematical Sciences; Springer: Cham, Switzerland, 2025; Volume 16. [Google Scholar]
- Brading, K.; Brown, H.R. Symmetries and Noether’s Theorems. In Symmetries in Physics: Philosophical Reflections; Brading, K., Castellani, E., Eds.; Cambridge University Press: Cambridge, UK, 2003; pp. 89–109. [Google Scholar]
- Atz, K.; Grisoni, F.; Schneider, G. Geometric deep learning on molecular representations. Nat. Mach. Intell. 2021, 3, 1023–1032. [Google Scholar] [CrossRef]
- Zhao, Y.; Siriwardane, E.M.D.; Wu, Z.; Hu, J. Physics guided deep learning for generative design of crystal materials with symmetry constraints. npj Comput. Mater. 2023, 9, 38. [Google Scholar] [CrossRef]
- Frenkel, N.; Fedorets, A.A.; Dombrovsky, L.A.; Nosonovsky, M.; Legchenkova, I.; Bormashenko, E. Continuous Symmetry Measure vs Voronoi Entropy of Droplet Clusters. J. Phys. Chem. C 2021, 125, 2431–2436. [Google Scholar] [CrossRef]
- Kreienkamp, K.L.; Klapp, S.H.L. Nonreciprocal Alignment Induces Asymmetric Clustering in Active Mixtures. Phys. Rev. Lett. 2024, 133, 258303. [Google Scholar] [CrossRef]
- Vermani, L.R. Elements of Algebraic Coding Theory; Routledge: New York, NY, USA, 2022. [Google Scholar]
- Kornyak, V.V. Discrete dynamical systems with symmetries: Computer analysis. Program. Comput. Soft. 2008, 34, 84–94. [Google Scholar] [CrossRef]
- Renner, R. Symmetry of large physical systems implies independence of subsystems. Nat. Phys. 2007, 3, 645–649. [Google Scholar] [CrossRef]
- Odintsov, S.D.; Oikonomou, V.K. Inverse symmetric inflationary attractors. Class. Quantum Grav. 2017, 34, 105009. [Google Scholar] [CrossRef]
- Hellmann, F.; Schultz, P.; Grabow, C.; Heitzig, J.; Kurths, J. Survivability of Deterministic Dynamical Systems. Sci. Rep. 2016, 6, 29654. [Google Scholar] [CrossRef]
- Pillay, A. Geometric Stability Theory; Oxford Academic: Oxford, UK, 1996. [Google Scholar]
- Goodsell, D.S.; Olson, A.J. Structural Symmetry and Protein Function. Ann. Rev. Biophys. 2000, 29, 105–153. [Google Scholar] [CrossRef] [PubMed]
- Bormashenko, E. Symmetry Breaking: One-Point Theorem. Symmetry 2025, 17, 1395. [Google Scholar] [CrossRef]
- Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J. 1948, 27, 379–423. [Google Scholar] [CrossRef]
- Ben-Naim, A. Entropy, Shannon’s Measure of Information and Boltzmann’s H-Theorem. Entropy 2017, 19, 48. [Google Scholar]
- Chatterjee, S.K. Crystallography and the World of Symmetry; Springer: Berlin, Germany, 2008; pp. 47–56. [Google Scholar]
- Albertson, M.; Collins, K. Symmetry breaking in graphs. Electron. J. Combin. 1996, 3, R18. [Google Scholar] [CrossRef]
- Xu, M.; Hu, W.; Han, Z.; Bai, H.; Deng, G.; Zhang, C. Symmetry-breaking dynamics of a flexible hub-beam system rotating around an eccentric axis. Mech. Syst. Signal Process. 2025, 222, 111757. [Google Scholar] [CrossRef]
- Zhu, H.; Han, Z.; Hu, W. Generalized Multi-Symplectic Analysis for Lateral Vibration of Vehicle–Bridge System Subjected to Wind Excitation. J. Vib. Eng. Technol. 2025, 13, 460. [Google Scholar] [CrossRef]
- Strocchi, F. Symmetry Breaking; Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 2005; Volume 732. [Google Scholar]
- Nambu, Y. Nobel lecture: Spontaneous symmetry breaking in particle physics: A case of cross fertilization. Rev. Mod. Phys. 2009, 81, 1015. [Google Scholar] [CrossRef]
- Carter, C.B.; Norton, M.G. Point Defects, Charge, and Diffusion. In Ceramic Materials; Springer: New York, NY, USA, 2013. [Google Scholar]
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Bormashenko, E. Probabilistic Measure of Symmetry Stability. Symmetry 2025, 17, 1675. https://doi.org/10.3390/sym17101675
Bormashenko E. Probabilistic Measure of Symmetry Stability. Symmetry. 2025; 17(10):1675. https://doi.org/10.3390/sym17101675
Chicago/Turabian StyleBormashenko, Edward. 2025. "Probabilistic Measure of Symmetry Stability" Symmetry 17, no. 10: 1675. https://doi.org/10.3390/sym17101675
APA StyleBormashenko, E. (2025). Probabilistic Measure of Symmetry Stability. Symmetry, 17(10), 1675. https://doi.org/10.3390/sym17101675